Problem Solving Bentuk Bilangan Riil

Seri Pemecahan Masalah

Jika pada bahasan sebelumnya kita bahas bilangan tidak nyata atau bilangan imajiner pada akar persamaan kuadrat, sekarang kita ketengahkan bahasan sebaliknya, yaitu akar nyta atau riil dari suatu persamaan kuadrat. 

Berikut permasalahannya

(sumber soal dari blog saya sendiri di wordpress)

$\begin{array}{rl}& \text{Akar riil terbesar untuk persamaan}\\ & {\displaystyle \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4}\\ & \text{adalah}p+\sqrt{q+\sqrt{r}}\text{dengan}p,q,\text{dan}r\text{adalah}\\ & \text{bilangan asli}.\text{Tentukanlah nilai}p+q+r\\ \\ & \mathbf{\text{Solusi}}:\end{array}$.

$\begin{array}{rl}& {\displaystyle \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4}\\ & \frac{3}{x-3}+1+\frac{5}{x-5}+1+\frac{17}{x-17}+1+\frac{19}{x-19}+1=x^2-11x\\ & \frac{3+(x-3)}{x-3}+\frac{5+(x-5)}{x-5}+\frac{17+(x-17)}{x-17}+\frac{19+(x-19)}{x-19}=x^2-11x\\ & \frac{x}{x-3}+\frac{x}{x-5}+\frac{x}{x-17}+\frac{x}{x-19}=x^2-11x\\ & \frac{x(x-19)+x(x-3)}{(x-3)(x-19)}+\frac{x(x-17)+x(x-5)}{(x-5)(x-17)}=x^2-11x\\ & \frac{2x^2-22x}{x^2-22x+57}+\frac{2x^2-22}{x^2-22x+85}=x^2-11x\\ & (x^2-11x)(\frac{2}{x^2-22x+57}+\frac{2}{x^2-22x+85})\\ & =x^2-11x,\text{misal}t=x^2-22x\\ & (\frac{2}{t+57}+\frac{2}{t+85})=\frac{x^2-11x}{x^2-11x}=1\\ & 2(t+85)+2(t+57)=(t+57)(t+85)\\ & 2t+170+2t+114=t^2+142t+4845\\ & 0=t^2+138t+4731\\ & t^2+138t+4731=0\{\begin{array}{c}a=1\\ b=138\\ c=4731\end{array}\\ & t_{1,2}={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}\\ & t_{1,2}={\displaystyle \frac{-138\pm \sqrt{{138}^2-4.1.4731}}{2}}\\ & ={\displaystyle \frac{-138\pm \sqrt{19044-18924}}{2}}\\ & ={\displaystyle \frac{-138\pm \sqrt{120}}{2}}\\ & ={\displaystyle \frac{-138\pm 2\sqrt{30}}{2}}\\ & =-69\pm \sqrt{30}\end{array}$.

$\begin{array}{rl}& \text{Selanjutnya}\\ & t_{1,2}=-69\pm \sqrt{30}\\ & x^2-22x=-69\pm \sqrt{30}\\ & x^2-22x+69\pm \sqrt{30}=0\\ & x^2-22x+69+\sqrt{30}=0\\ & \text{atau}{x}^2-22x+69-\sqrt{30}\\ & \\ & \text{dengan cara yang}\text{semisal diatas}\\ & \\ & x_{1,2}={\displaystyle \frac{22\pm \sqrt{{22}^2-4(69+\sqrt{30})}}{2}}\\ & \text{atau}{x}_{3,4}={\displaystyle \frac{22\pm \sqrt{{22}^2-4(69-\sqrt{30})}}{2}}\\ & x_{1,2}={\displaystyle \frac{22\pm \sqrt{484-276-4\sqrt{30}}}{2}}\\ & \text{atau}{x}_{3,4}={\displaystyle \frac{22\pm \sqrt{484-276+4\sqrt{30}}}{2}}\\ & x_{1,2}={\displaystyle \frac{22\pm \sqrt{208-4\sqrt{30}}}{2}}\\ & \text{atau}{x}_{3,4}={\displaystyle \frac{22\pm \sqrt{208+4\sqrt{30}}}{2}}\\ & x_{1,2}={\displaystyle \frac{22\pm 2\sqrt{52-\sqrt{30}}}{2}}\\ & \text{atau}{x}_{3,4}={\displaystyle \frac{22\pm 2\sqrt{52+\sqrt{30}}}{2}}\\ & x_{1,2}=11\pm \sqrt{52-\sqrt{30}}\\ & \text{atau}{x}_{3,4}=11\pm \sqrt{52+\sqrt{30}}\\ & \\ & \text{Maka},\\ & \{\begin{array}{c}{x}_1=11+\sqrt{52-\sqrt{30}}\\ \\ x_2=11-\sqrt{52-\sqrt{30}}\end{array}\\ & \text{atau}\{\begin{array}{c}{x}_3=11+\sqrt{52+\sqrt{30}}\\ \\ x_4=11-\sqrt{52+\sqrt{30}}\end{array}\end{array}$.

$\begin{array}{rl}\text{Selanjutnya nilai}& \text{yang paling pas sesuai soal adalah}:\\ & x_3=11+\sqrt{52+\sqrt{30}}=p+\sqrt{q+\sqrt{r}}\\ \text{Sehingga nilai}& p+q+r=11+52+30=93\end{array}$.

Problem Solving Bentuk Bilangan Imajiner (Bilangan Tidak Nyata)

Seri Pemecahan Masalah

Suatu ketika saya sharing-sharing mengenai soal bentuk perpangkatan dari salah seorang teman yang kebetulan memang soalnya membuat penasayaran untuk ditemukan jawabannya.

Berikut soalnya

Saat saya melihat soalnya dengan pangkat berupa angka yang seolah berpola tapi agak susah dicari hungan antara keduanya. Yang satu bilangan utuh yang satu lagi bentuk pecahan (bilangan pada soal, bukan pada yang diketahui). Tapi ada sedikit petunjuk yang mensiratkan soal di atas akan segera dapat dipecahkankan, yaitu posisi yang diketahui  $x+{\displaystyle \frac{1}{x}=-1}$ adalah salah satu bentuk persamaan kuadrat dengan akar kemungkinan rasional atau imajiner/khayal/tidak nyata dan pangkat pada soal yang semuanya menunjukkan kelipatan 3, yaitu 1234567891011 dan yang satunya posisi penyebut dengan pangkat 1110987654321 dengan basis/bilangan pokok perpangkatannya sama dengan yang diketahui dari soal yaitu  $a$.

Sebelumnya saya pernah menyinggung mengenai istilah definit positif dan definit negatif (silahkan klik di sini) yang kurang lebih istilah tersebut sangat berkaitan dengan akar persamaan kuadrat yang berbentuk imajiner.

Ok, kita kembali ke arah penyelesaian soal di atas, yaitu:

$\begin{array}{rl}& a+{\displaystyle \frac{1}{a}=-1\iff a^2+1=-a}\\ & \iff a^2+a+1=0\\ & \iff a_{1,2}={\displaystyle \frac{-1\pm \sqrt{-3}}{2}={\displaystyle \frac{-1\pm \sqrt{3.(-1)}}{2}}}\\ & ={\displaystyle \frac{-1\pm \sqrt{3}\sqrt{-1}}{2}=\frac{-1\pm \sqrt{3}i}{2}}\\ & \text{dengan}i=\sqrt{-1}\end{array}$.

$\begin{array}{rl}& \text{Misalkan kita pilih}a={\displaystyle \frac{-1+\sqrt{3}i}{2}}\\ & \text{maka nilai dari}\\ & {\displaystyle \frac{1}{a}={\displaystyle \frac{1}{{\displaystyle \frac{-1+\sqrt{3}i}{2}}}={\displaystyle \frac{2}{-1+\sqrt{3}i}={\displaystyle \frac{2}{\sqrt{3}i-1}}}}}\\ & ={\displaystyle \frac{2}{\sqrt{3}i-1}\times {\displaystyle \frac{\sqrt{3}i+1}{\sqrt{3}i+1}={\displaystyle \frac{2(\sqrt{3}i+1)}{-3-1}}}}\\ & =-{\displaystyle \frac{2(\sqrt{3}i+1)}{-4}={\displaystyle \frac{\sqrt{3}i+1}{-2}\text{atau}}}\\ & {\displaystyle \frac{1}{a}={\displaystyle \frac{-1-\sqrt{3}i}{2}}}\end{array}$.

Penjabaran bentuk pangkat dari salah satu akar ternyata membentuk pola yang unik sebagaimana bentuk berikut:

$\begin{array}{rl}& \{\begin{array}{ll}a& ={\displaystyle \frac{-1+\sqrt{3}i}{2}}\\ {\displaystyle \frac{1}{a}}& ={\displaystyle \frac{-1-\sqrt{3}i}{2}}\end{array},\{\begin{array}{ll}{a}^2& ={\displaystyle \frac{-1-\sqrt{3}i}{2}}\\ {\displaystyle \frac{1}{a^2}}& ={\displaystyle \frac{-1+\sqrt{3}i}{2}}\end{array}\\ & \{\begin{array}{ll}{a}^3& =1\\ {\displaystyle \frac{1}{a^3}}& =1\end{array}\\ & \{\begin{array}{ll}{a}^4& ={\displaystyle \frac{-1+\sqrt{3}i}{2}}\\ {\displaystyle \frac{1}{a^4}}& ={\displaystyle \frac{-1-\sqrt{3}i}{2}}\end{array}\{\begin{array}{ll}{a}^5& ={\displaystyle \frac{-1-\sqrt{3}i}{2}}\\ {\displaystyle \frac{1}{a^5}}& ={\displaystyle \frac{-1+\sqrt{3}i}{2}}\end{array}\\ & \{\begin{array}{ll}{a}^6& =1\\ {\displaystyle \frac{1}{a^6}}& =1\end{array}\\ & ⋮\\ & \cdots \cdots \{\begin{array}{ll}{a}^9& =1\\ {\displaystyle \frac{1}{a^9}}& =1\end{array}\\ & \cdots \cdots \{\begin{array}{ll}{a}^{12}& =1\\ {\displaystyle \frac{1}{a^{12}}}& =1\end{array}\\ & \cdots \cdots \{\begin{array}{ll}{a}^{15}& =1\\ {\displaystyle \frac{1}{a^{15}}}& =1\end{array}\\ & \text{dan seterusnya}\\ & \end{array}$.

Jadi, setiap pangkat kelipatan 3 ternyata sama dengan 1, sehingga ini mengakibatkan soal di atas dapat dituliskan lagi dengan

$\begin{array}{rl}& \text{Perhatikan lagi bentuk soal}\\ & a^{1234567891011}+{\displaystyle \frac{1}{a^{11100987654321}}}\\ & =a^{3m}+{\displaystyle \frac{1}{a^{3n}}=1+{\displaystyle \frac{1}{1}=1+1=2}}\end{array}$.

Contoh Soal 13 Statistika

$\begin{array}{ll}56.& \text{Simpangan baku dari data berikut}:\\ & 6,7,4,5,3\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& {\displaystyle \frac{1}{2}}& & & \text{d}.& \sqrt{2}\\ \text{b}.& {\displaystyle \frac{1}{2}\sqrt{2}}& \text{c}.& {\displaystyle \frac{1}{2}\sqrt{3}}& \text{e}.& \sqrt{3}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 6,7,4,5,3\\ & \text{Simpangan bakuya adalah}:\\ & S=\sqrt{S^2}=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{6+7+4+5+3}{5}={\displaystyle \frac{25}{5}=5}}\\ & \text{maka nilai}\\ & S=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & =\sqrt{{\displaystyle \frac{1}{5}((6-5{)}^2+(7-5{)}^2+(4-5{)}^2+(5-5{)}^2+(3-5{)}^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(1^2+2^2+1^2+0+2^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(1+4+1+4)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(10)}}=\sqrt{{\displaystyle \frac{10}{5}}}=\sqrt{2}\end{array}\end{array}$.

$\begin{array}{ll}57.& \mathbf{\text{UN 2010}}\\ & \text{Simpangan baku dari data berikut}:\\ & 2,3,4,5,6\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& \sqrt{15}& & & \text{d}.& \sqrt{3}\\ \text{b}.& {\displaystyle \sqrt{10}}& \text{c}.& {\displaystyle \sqrt{5}}& \text{e}.& \sqrt{2}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 2,3,4,5,6\\ & \text{Simpangan bakuya adalah}:\\ & S=\sqrt{S^2}=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{2+3+4+5+6}{5}={\displaystyle \frac{20}{5}=4}}\\ & \text{maka nilai}\\ & S=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & =\sqrt{{\displaystyle \frac{1}{5}((2-5{)}^2+(3-5{)}^2+(4-5{)}^2+(5-5{)}^2+(6-5{)}^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(3^2+2^2+1^2+0+1^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(9+4+1+1)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(15)}}=\sqrt{{\displaystyle \frac{15}{5}}}=\sqrt{3}\end{array}\end{array}$.

$\begin{array}{ll}58.& \text{Simpangan baku dari data berikut}:\\ & 7,9,11,13,15\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& 2,4& & & \text{d}.& 2,8\\ \text{b}.& {\displaystyle 2,5}& \text{c}.& {\displaystyle 2,7}& \text{e}.& 2,9\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 7,9,11,13,15\\ & \text{Simpangan bakuya adalah}:\\ & S=\sqrt{S^2}=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{7+9+11+13+15}{5}={\displaystyle \frac{55}{5}=11}}\\ & \text{maka nilai}\\ & S=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & =\sqrt{{\displaystyle \frac{1}{5}((7-11{)}^2+(9-11{)}^2+(11-11{)}^2+(13-11{)}^2+(15-11{)}^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(4^2+2^2+0+2^2+4^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(16+4+4+16)}}\\ & =\sqrt{{\displaystyle \frac{1}{5}(40)}}=\sqrt{{\displaystyle \frac{40}{5}}}=\sqrt{8}=2,82..\end{array}\end{array}$.

 $\begin{array}{ll}59.& \text{Simpangan baku dari data berikut}:\\ & 2,4,4,5,6,6,7,8,9,9\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& 4\sqrt{3}& & & \text{d}.& {\displaystyle \frac{2}{5}\sqrt{30}}\\ \text{b}.& 2{\displaystyle \frac{2}{5}}& \text{c}.& {\displaystyle \sqrt{5}}& \text{e}.& 2\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 2,4,4,5,6,6,7,8,9,9\\ & \text{Simpangan bakunya adalah}:\\ & S=\sqrt{S^2}=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{2+4+4+5+6+6+7+8+9+9}{10}={\displaystyle \frac{60}{10}=6}}\\ & \text{maka nilai}\\ & S=\sqrt{{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}}\\ & =\sqrt{{\displaystyle \frac{1}{10}((2-6{)}^2+2(4-6{)}^2+(5-6{)}^2+2(6-6{)}^2+(7-6{)}^2+(8-6{)}^2+2(9-6{)}^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{10}(4^2+{2.2}^2+1^2+0+1^2+2^2+{2.3}^2)}}\\ & =\sqrt{{\displaystyle \frac{1}{10}(16+8+1+1+4+18)}}\\ & =\sqrt{{\displaystyle \frac{1}{10}(48)}}=\sqrt{{\displaystyle \frac{48}{10}}}=\sqrt{{\displaystyle \frac{120}{25}}}={\displaystyle \frac{2}{5}\sqrt{30}}\end{array}\end{array}$.

Contoh Soal 12 Statistika

$\begin{array}{ll}51.& \text{Simpangan kuartil dari data}\\ & 56a378\text{adalah}1{\displaystyle \frac{1}{2}.\text{Jika median datanya}}\\ & \text{adalah}5{\displaystyle \frac{1}{2},\text{maka rata-rata data}\text{tersebut adalah}....}\\ & \begin{array}{llllll}\text{a}.& \text{4}& & & \text{d}.& \text{5}{\displaystyle \frac{1}{2}}\\ \text{b}.& \text{4}{\displaystyle \frac{1}{2}}& \text{c}.& \text{5}& \text{e}.& \text{6}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 56a378\Rightarrow n=6\\ & \text{karena mediannya}=M_e=Q_2=5\frac{1}{2}=\frac{11}{2}\\ & \text{ data menjadi}:a35678\text{atau}3a5678\\ & \text{Simpangan kuartilnya adalah}1{\displaystyle \frac{1}{2}={\displaystyle \frac{3}{2}}}\\ & Q_d={\displaystyle \frac{1}{2}(Q_3-Q_1)={\displaystyle \frac{3}{2}\Rightarrow Q_3-Q_1=3}}\\ & \text{maka}\\ & 3=Q_3-Q_1=x_{{.}_{\frac{3}{4}n+\frac{1}{2}}}-x_{{.}_{\frac{1}{4}n+\frac{1}{2}}}=x_{{.}_{\frac{3}{4}6+\frac{1}{2}}}-x_{{.}_{\frac{1}{4}6+\frac{1}{2}}}\\ & =(x_{{.}_5}-x_{{.}_2})=7-x_{{.}_2}=3\Rightarrow x_{{.}_2}=4\\ & \text{Jadi, rata-ratanya adalah}:\\ & \bar{x}=\frac{3+4+5+6+7+8}{6}=5,5\end{array}\end{array}$.

$\begin{array}{ll}52.& \text{Simpangan kuartil dari data}\\ & \text{berikut}\\ & 61,61,50,50,53,53,70,61\\ & 53,70,53,61,50,61,70\\ & \text{adalah}....\\ & \begin{array}{llllll}\text{a}.& \text{10}& & & \text{d}.& \text{6}\\ \text{b}.& \text{9}& \text{c}.& \text{8}& \text{e}.& \text{5}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{-}\\ & \begin{array}{rl}& \text{Data mula-mula}n=15\\ & 61,61,50,50,53,53,70,61\\ & 53,70,53,61,50,61,70\\ & \text{data durutkan}\\ & 50,50,50,53,53,53,53\\ & 61,61,61,61,61,70,70,70\\ & \text{Simpangan kuartil adalah}{Q}_d,\\ & Q_d={\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ & ={\displaystyle \frac{1}{2}(x_{{.}_{\frac{3}{4}(n+1)}}-x_{{.}_{\frac{1}{4}(n+1)}})}\\ & ={\displaystyle \frac{1}{2}(x_{{.}_{\frac{3}{4}(15+1)}}-x_{{.}_{\frac{1}{4}(15+1)}})}\\ & ={\displaystyle \frac{1}{2}(x_{{.}_{12}}-x_{{.}_4})}\\ & ={\displaystyle \frac{1}{2}(61-53)={\displaystyle \frac{1}{2}\times 8=4}}\\ & \text{Jadi},Q_d=4\end{array}\end{array}$.

$\begin{array}{ll}53.& \text{Simpangan rata-rata dari data berikut}:\\ & 642810\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& \text{2}& & & \text{d}.& \text{3},0\\ \text{b}.& \text{2},4& \text{c}.& \text{2},5& \text{e}.& \text{3},5\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{b}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 642810\\ & \text{Simpangan rata-ratanya adalah}:\\ & SR={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{6+4+2+8+10}{5}={\displaystyle \frac{30}{5}=6}}\\ & \text{maka nilai}\\ & SR={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & ={\displaystyle \frac{1}{5}(|6-6|+|4-6|+|2-6|+|8-6|+|10-6|)}\\ & ={\displaystyle \frac{1}{5}(\left|0\right|+|-2|+|-4|+\left|2\right|+\left|4\right|)}\\ & ={\displaystyle \frac{1}{5}(0+2+4+2+4)}\\ & ={\displaystyle \frac{1}{5}(12)={\displaystyle \frac{12}{5}=2,4}}\end{array}\end{array}$.

$\begin{array}{ll}54.& \text{Simpangan rata-rata dari data berikut}:\\ & 10,8,7,10,7,5,8,6,10,9\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& \text{1,0}& & & \text{d}.& \text{8,0}\\ \text{b}.& \text{1,4}& \text{c}.& \text{6,0}& \text{e}.& \text{14,0}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{b}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 10,8,7,10,7,5,8,6,10,9\\ & \text{Simpangan rata-ratanya adalah}:\\ & SR={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{10+8+7+10+7+5+8+6+10+9}{10}}\\ & ={\displaystyle \frac{80}{10}=8}\\ & \text{maka nilai}\\ & SR={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & ={\displaystyle \frac{1}{10}(|5-8|+|6-8|+2|7-8|+2|8-8|+|9-8|+3|10-8|)}\\ & ={\displaystyle \frac{1}{10}(|-3|+|-2|+2|-1|+2\left|0\right|+\left|1\right|+3\left|2\right|)}\\ & ={\displaystyle \frac{1}{10}(3+2+2+0+1+6)}\\ & ={\displaystyle \frac{1}{10}(14)={\displaystyle \frac{14}{10}=1,4}}\end{array}\end{array}$.

$\begin{array}{ll}55.& \text{Nilai variansi dari data}\\ & 6,7,7,8,8,8,8,12\text{ adalah}....\\ \\ & \begin{array}{llllll}\text{a}.& \text{1}& & & \text{d}.& \text{8}\\ \text{b}.& {\displaystyle \frac{26}{8}}& \text{c}.& {\displaystyle \frac{11}{4}}& \text{e}.& \text{22}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{c}\\ & \begin{array}{rl}& \text{Diketahui data sebagai berikut}\\ & 6,7,7,8,8,8,8,12\\ & \text{Variannya adalah}:\\ & S^2={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}\\ & \text{dengan}\\ & \bar{x}={\displaystyle \frac{6+7+7+8+8+8+8+12}{8}={\displaystyle \frac{64}{8}=8}}\\ & \text{maka nilai}\\ & S^2={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}\\ & ={\displaystyle \frac{1}{8}((6-8{)}^2+2(7-6{)}^2+4(8-8{)}^2+(12-8{)}^2)}\\ & ={\displaystyle \frac{1}{8}(2^2+2.1+4.0+4^2)}\\ & ={\displaystyle \frac{1}{8}(4+2+0+16)}\\ & ={\displaystyle \frac{1}{8}(22)={\displaystyle \frac{11}{4}=2,75}}\end{array}\end{array}$.

Contoh Soal 11 Statistika

$\begin{array}{ll}46.& \text{Rata-rata dari data yang disajikan }\\ & \text{dengan hitogram berikut adalah}....\end{array}$.

$.\begin{array}{ll}& \begin{array}{l}\\ \text{a}.& 41,372& & & \text{d}.& 43,135\\ \text{b}.& 42,150& \text{c}.& 43,125& \text{e}.& 44,250\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \begin{array}{|cccc|}\hline \begin{array}{rl}& \text{Berat}\\ & \text{Badan}\end{array}& {\text{f}}_1& \begin{array}{rl}& \text{Nilai Tengah}\\ & ({\text{x}}_1)\end{array}& {\text{f}}_1{\text{x}}_1\\ 30-34& 5& 32& 160\\ 35-39& 7& 37& 259\\ 40-44& 12& 42& 504\\ 45-49& 9& 47& 423\\ 50-54& 4& 52& 208\\ 55-59& 3& 57& 171\\ & \sum {\text{f}}_1=40& & \sum {\text{f}}_1{\text{x}}_1=1725\\ \hline\end{array}\\ & \text{maka nilai dari}\bar{\text{x}}\text{adalah}:\\ & \bar{\text{x}}={\displaystyle \frac{\sum {\text{f}}_1{\text{x}}_1}{\sum {\text{f}}_1}={\displaystyle \frac{1725}{40}=43,125}}\end{array}\end{array}$.

$\begin{array}{ll}47.& (\mathbf{\text{UN Mat IPA 2006}})\\ & \text{Perhatikan gambar berikut}\end{array}$.

$.\begin{array}{ll}& \text{Berat badan pada suatu kelas tersaji dengan}\\ & \text{bentuk histogram seperti pada gambar di atas}\\ & \text{Rata-rata berat badan tersebut adalah}....\text{Kg}\\ & \begin{array}{l}\\ \text{a}.& 64,5& & & \text{d}.& 66\\ \text{b}.& 65& \text{c}.& 65,5& \text{e}.& 66,5\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{b}}\\ & \begin{array}{rl}& \begin{array}{|cccc|}\hline \begin{array}{rl}& \text{Berat}\\ & \text{Badan}\end{array}& {\text{f}}_1& \begin{array}{rl}& \text{Nilai Tengah}\\ & ({\text{x}}_1)\end{array}& {\text{f}}_1{\text{x}}_1\\ 50-54& 4& 52& 208\\ 55-59& 6& 57& 342\\ 60-64& 8& 62& 496\\ 65-69& 10& 67& 670\\ 70-74& 8& 72& 576\\ 75-79& 4& 77& 308\\ & \sum {\text{f}}_1=40& & \sum {\text{f}}_1{\text{x}}_1=2600\\ \hline\end{array}\\ & \text{maka nilai dari}\bar{\text{x}}\text{adalah}:\\ & \bar{\text{x}}={\displaystyle \frac{\sum {\text{f}}_1{\text{x}}_1}{\sum {\text{f}}_1}={\displaystyle \frac{2600}{40}=65}}\end{array}\end{array}$.

$\begin{array}{ll}48.& \text{Diketahui tabel distribusi frekuensi berikut}\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 145-149& 3\\ 150-154& 5\\ 155-159& 17\\ 160-164& 15\\ 165-169& 8\\ 170-174& 2\\ \hline\end{array}\\ & \text{Kuartil bawah dapat dinyatakan dengan}....\\ & \begin{array}{l}\\ \text{a}.& 149,5+\left({\displaystyle \frac{12,5-3}{8}}\right).5\\ \text{b}.& 150+\left({\displaystyle \frac{12,5-3}{8}}\right).5\\ \text{c}.& 155+\left({\displaystyle \frac{12,5-8}{17}}\right).5\\ \text{d}.& 154,5+\left({\displaystyle \frac{12,5-8}{17}}\right).5\\ \text{e}.& 155,5+\left({\displaystyle \frac{12,5-8}{17}}\right).5\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \text{Diketahui}n=50\\ & \text{Ditanyakan kuartil bawah, maka hal ini}\\ & =Q_1\Rightarrow x_{{.}_{\frac{1}{4}n}}=x_{{.}_{\frac{1}{4}50}}=x_{{.}_{12,5}}\\ & \text{Perhatikan tabelnya lagi}\\ & \text{dengan penambahan warna berikut ini}\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 145-149& 3\\ 150-154& 5\\ 155-159& 17\\ 160-164& 15\\ 165-169& 8\\ 170-174& 2\\ \hline\end{array}\\ & \text{maka nilai dari}{Q}_1\text{adalahh}:\\ & Q_1=L+\left({\displaystyle \frac{\frac{1}{4}n-f_k}{f}}\right).c\\ & Q_1=154,5+\left({\displaystyle \frac{12,5-8}{17}}\right).5\end{array}\end{array}$.

$\begin{array}{ll}49.& \text{Diketahui tabel distribusi frekuensi berikut}\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 5-9& 2\\ 10-14& 8\\ 15-19& 10\\ 20-24& 7\\ 25-29& 3\\ \hline\end{array}\\ & \text{Median dari tabel di atas adalah}....\\ & \begin{array}{l}\\ \text{a}.& 15,0& & & \text{d}.& 16,5\\ \text{b}.& 15,5& \text{c}.& 16,0& \text{e}.& 17,0\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{e}\\ & \begin{array}{rl}& \text{Diketahui}n=30\\ & \text{Ditanyakan median, maka formulanya}\\ & =Q_2\Rightarrow x_{{.}_{\frac{2}{4}n}}=x_{{.}_{\frac{1}{2}.30}}=x_{{.}_{15}}\\ & \text{Perhatikan tabelnya lagi}\\ & \text{dengan penambahan warna berikut ini}\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 5-9& 2\\ 10-14& 8\\ 15-19& 10\\ 20-24& 7\\ 25-29& 3\\ \hline\end{array}\\ & \text{maka nilai dari}{Q}_2\text{adalah}:\\ & Q_2=L+\left({\displaystyle \frac{\frac{2}{4}n-f_k}{f}}\right).c\\ & Q_1=14,5+\left({\displaystyle \frac{15-10}{10}}\right).5\\ & =14,5+\frac{25}{10}=14,5+2,5=17,0\end{array}\end{array}$.

$\begin{array}{ll}50.& \text{Jangkauan antarkuartil dari data}\\ & \text{berikut}:\\ & 3625564055424364\\ & 82702835384554\\ & \text{adalah}....\\ & \begin{array}{llllll}\text{a}.& \text{20}& & & \text{d}.& \text{5}\\ \text{b}.& \text{10}& \text{c}.& \text{8}& \text{e}.& \text{3}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{a}\\ & \begin{array}{rl}& \text{Data mula-mula}n=15\\ & 3625564055424364\\ & 82702835384554\\ & \text{data durutkan}\\ & 2528353638404243\\ & 45545556647082\\ & \text{Jangkauan antarkuartil adalah}H,\\ & H=Q_3-Q_1\\ & =x_{{.}_{\frac{3}{4}(n+1)}}-x_{{.}_{\frac{1}{4}(n+1)}}\\ & =x_{{.}_{\frac{3}{4}(15+1)}}-x_{{.}_{\frac{1}{4}(15+1)}}\\ & =(x_{{.}_{12}}-x_{{.}_4})\\ & =56-36\\ & \text{Jadi},H=Q_3-Q_1=20\end{array}\end{array}$.

Contoh Soal 10 Statistika

Soal sebelumnya (yaitu Contoh Soal 9 Statistika) klik di sini

$\begin{array}{ll}41.& \text{Median dan modus dari data berikut}\\ & 3,6,7,8,4,5,9,6\\ & \text{ adalah}....\\ & \begin{array}{llllll}\text{a}.& \text{7 dan 5}& & & \text{d}.& \text{5 dan 6}\frac{1}{2}\\ \text{b}.& \text{6 dan 6}& \text{c}.& \text{6 dan 7}& \text{e}.& \text{5 dan 6}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{b}\\ & \begin{array}{rl}& \text{Diketahui}n=8\\ & \text{Datum diurutkan dari kecil ke besar}\\ & :3,4,5,6,6,7,8,9\\ & \bullet \mathbf{\text{modus}}=M_o=6,\\ & \bullet \mathbf{\text{mean}}=\bar{x}=6\end{array}\end{array}$.

$\begin{array}{ll}42.& \text{Hasil tes matematika di suatu kelas yang}\\ & \text{diikuti tes 49 siswa menghasilkan nilai}\\ & \text{rata-rata 7}.\text{Jika Andi ikut tes susulan}\\ & 7,04.\text{Nilai Andi adalah}....\\ & \begin{array}{llllll}\text{a}.& \text{7,5}& & & \text{d}.& \text{9}\\ \text{b}.& \text{8}& \text{c}.& \text{8,5}& \text{e}.& \text{9,5}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}\text{Misalkan}& y=\text{besar nilai Andi}\\ {\bar{x}}_{gabungan}& ={\displaystyle \frac{n.\bar{x}+y}{n+1}}\\ 7,04& ={\displaystyle \frac{49\times 7+y}{49+1}}\\ 7,04& ={\displaystyle \frac{343+y}{50}}\\ 343+y& =50\times (7,04)\\ 343+y& =352\\ y& =352-343\\ & =9\end{array}\end{array}$.

$\begin{array}{ll}43.& \text{Mean dari tabel berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 50-54& 4\\ 55-59& 6\\ 60-64& 10\\ \hline\end{array}\\ & \begin{array}{llllll}\text{a}.& \text{60,5}& & & \text{d}.& \text{58,5}\\ \text{b}.& \text{60}& \text{c}.& \text{59,5}& \text{e}.& \text{57}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{d}\\ & \begin{array}{rl}& \begin{array}{|ccccc|}\hline \text{Ukuran}& x_i& d_i=x_i-x_s& f_i& f_i\times d_i\\ 50-54& 52& -10& 4& -40\\ 55-59& 57& -5& 6& -30\\ 60-64& 62& 0& 10& 0\\ \text{Jumlah}& & & 20& -70\\ \hline\end{array}\\ & \bar{x}=x_s+{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{f}_i\times d_1}}{{\displaystyle \sum _{i=0}^n{f}_i}}=62+{\displaystyle \frac{-70}{20}}}\\ & =62-3,5=58,5\end{array}\end{array}$.

$\begin{array}{ll}44.& \text{Median dari tabel berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 47-49& 1\\ 50-52& 6\\ 53-55& 6\\ 56-58& 7\\ 59-61& 4\\ \hline\end{array}\\ & \begin{array}{llllll}\text{a}.& \text{55,5}& & & \text{d}.& \text{53,5}\\ \text{b}.& \text{55}& \text{c}.& \text{54,5}& \text{e}.& \text{53}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{b}\\ & \begin{array}{rl}& \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 47-49& 1\\ 50-52& 6\\ 53-55& 6\\ 56-58& 7\\ 59-61& 4\\ & 24\\ \hline\end{array}\\ & \text{Median posisi datumnya}:\\ & \text{datum ke}-\left({\displaystyle \frac{24}{2}}\right)=x_{{.}_{12}}\\ & \text{dan terletak di interval}\\ & 53-55,\text{dengan}{f}_k=1+6=7\\ & \text{serta}c=3\\ & M_e=Q_2=L_i+c\left({\displaystyle \frac{{\displaystyle \frac{n}{2}-f_k}}{f}}\right)\\ & M_e=Q_2=52,2+3\left({\displaystyle \frac{{\displaystyle \frac{24}{2}-7}}{6}}\right)\\ & =52,5+3\left({\displaystyle \frac{12-7}{6}}\right)\\ & =52,5+\left({\displaystyle \frac{5}{2}}\right)\\ & =52,5+2,5\\ & =55\end{array}\end{array}$.

$\begin{array}{ll}45.& \text{Modus dari tabel berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 50-54& 4\\ 55-59& 6\\ 60-64& 8\\ 65-69& 16\\ 70-74& 10\\ 75-79& 4\\ 80-84& 2\\ \hline\end{array}\\ & \begin{array}{l}\\ \text{a}.& \text{67,32}& & & \text{d}.& \text{70,12}\\ \text{b}.& \text{67,36}& \text{c}.& \text{67,56}& \text{e}.& \text{70,36}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\text{b}\\ & \begin{array}{rl}& \text{Perhatikan tabelnya lagi}\\ & \text{dengan penambahan warna berikut ini}\\ & \begin{array}{|cc|}\hline \text{Ukuran}& \text{Frekuensi}\\ 50-54& 4\\ 55-59& 6\\ 60-64& 8\\ 65-69& 16\\ 70-74& 10\\ 75-79& 4\\ 80-84& 2\\ \hline\end{array}\\ & \text{Diketahui}n=50\\ & \text{modus terletak pada kelas}\\ & \text{interval dengan frekuensi}\\ & \text{terbesar, yaitu}:16.\text{Kelas intervalnya}\\ & 65-69,\text{dengan}c=5\\ & \text{serta}\{\begin{array}{ll}{\mathrm{△}}_1& =f-f_1=16-8=8\\ {\mathrm{△}}_2& =f-f_2=16-10=6\end{array}\\ & M_o=L+c\left({\displaystyle \frac{{\mathrm{△}}_1}{{\mathrm{△}}_1+{\mathrm{△}}_2}}\right)\\ & =64,5+5\left({\displaystyle \frac{8}{8+6}}\right)\\ & =64,5+{\displaystyle \frac{40}{14}}\\ & =64,5+2,857\\ & =67,36\end{array}\end{array}$.

Koefisien Keragaman (Koefisien Variansi)

A. Pengertian

Pada bahasan ini untuk membandingkan dua atau lebih distribusi data yang sejenis dapat digunakan koefisien keragaman. Koefisien variansi adalah nilai dari standar deviasi suatu data dibagi dengan rata-ratanya.

B. Formula koefisien Variansi

Jika diketahui  $S$ adalah simpangan baku dan  $\bar{x}$ adalah rataan hitung suatu data, maka koefidien variansinya (V) dirumuskan dengan:

$V={\displaystyle \frac{S}{\bar{x}}\times 100}$.

$\text{CONTOH SOAL}$.

Contoh 1

Coba perhatikan lagi data pada halaman ini di sini, dengan datanya adalah:

$\begin{array}{|cccccc|}\hline \text{Nilai}& 47-49& 50-52& 53-55& 56-58& 59-61\\ \text{Frek}& 2& 4& 6& 5& 3\\ \hline\end{array}$. 

Dari perhitungan untuk data tersebut didapatkan besar rataan hitungnya adalah 54,45 dan simpangan bakunya adalah 3,58, maka koefisien dari variansi dari data tersebut adalah:

$\begin{array}{rl}V& ={\displaystyle \frac{S}{\bar{x}}\times 100\mathrm{\%}}\\ & ={\displaystyle \frac{3,58}{54,45}\times 100\mathrm{\%}}\\ & =6,57\mathrm{\%}\end{array}$.

Contoh 2

Diketahui nilai ulangan matematika suatu kelas di suatu waktu memiliki rataan 78 dengan simpangan bakunya adalah 7, sedangkan untuk nilai ulangan kimia dari kelas tersebut mendapatkan rataan 62 dan simpangan bakunya adalah 6. Tentukanlah mata pelajaran mana dari keduanya yang telah diuhikan itu yang memiliki penyebaran data yang lebih kecil

Jawab:

Dari data di atas, jika kita hanya berpatokan pada hasil simpangan baku kedua mapel yang telah diujikan tersebut tentunya mapel kimia akan memiliki persebaran yang lebih kecil dari pada mapel matematika. Akan tetapi adalah perhitungan yang lebih baik tentang permasalahan di atas, yaitu dengan menggunkan rumus koefisien variansi sebagaimana perhitungan berikut ini:

$\begin{array}{|cc|}\hline \text{Mapel Matematika}& \text{Mapel Kimia}\\ \begin{array}{rl}V& ={\displaystyle \frac{7}{78}\times 100\mathrm{\%}}\\ & =8,97\mathrm{\%}\end{array}& \begin{array}{rl}V& ={\displaystyle \frac{6}{62}\times 100\mathrm{\%}}\\ & =9,68\mathrm{\%}\end{array}\\ \hline\end{array}$

Tampak dari perhitungan koefisien variansi di atas bahwa nilai ulangan mapel matematika memiliki sebaran relatif lebih kecil dari pada hasil ulangan mapel kimia.

C. Angka Baku

Misalkan ada suatu permasalahan seorang siswa saat ulangan matematika mendapatkan nilai 8 di mana rataan kelasnya adalah 6,5 dan simpangan bakunya adalah 2. Sedangkan untuk hasil ulangan kimia ia berhasil mendapatkan nilai 9 yang rataan kelasnya 7,5 dan simpangan bakunya 3. Pertanyaannnya adalah hasil yang didapatkan anak tersebut kedudukannya mana yang lebih baik?

Untuk menjawab pertanyaan di atas kita dapat menggunkan angka baku, yaitu  $z={\displaystyle \frac{x-\bar{x}}{S}}$.

Berdasarkan nilai kita bisa tentukan angka baku nilai siswa tersebut, yaitu:

$\begin{array}{rl}\text{matematika}:z& ={\displaystyle \frac{8-6,5}{2}=0,75}\\ \text{fisika}:z& ={\displaystyle \frac{9-7,2}{3}=0,60}\end{array}$.

Dari perhitungan angka bakunya, tampak bahwa nilai ulangan matematika siswa tersebut lebih besar dari angka baku fisikanya. Hal ini menunjukkan nilai matematika siswa tersebut adalah yang lebih baik.

Ukuran Penyebaran Data Berkelompok (Materi Kelas XII Matematika Wajib) (Bagian 2)

 B. 2 Data Berkelompok

$\begin{array}{|cll|}\hline \text{No}& \text{Data Dispersi}& \text{Keterangan}\\ 1.& \text{Jangkauan}& \begin{array}{rl}\text{a}.& \text{selisih titik tengah}\\ & \text{kelas tertinggi dengan}\\ & \text{titik tengah kelas}\\ & \text{terendah}\\ \text{b}.& \text{selisih tepi atas kelas}\\ & \text{kelas tertinggi dengan}\\ & \text{tepi bawah kelas}\\ & \text{terendah}\end{array}\\ 2.& H& Q_3-Q_1\\ 3.& Q_d& {\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ 4.& SR& {\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ 5.& S^2& {\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i{(x_i-\bar{x})}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ 6.& S& \sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i{(x_i-\bar{x})}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}}\\ \hline\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukanlah nilai simpangan rata-ratanya}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& \text{47-49}& \text{50-52}& \text{53-55}& \text{56-58}& \text{59-61}\\ \text{Frek}& 2& 4& 6& 5& 3\\ \hline\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \mathbf{\text{Alternatif 1}}\\ & \begin{array}{ll}& \text{Perhatikan tabel berikut}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& x_i& f_i& f_i.x_i& |x_i-\bar{x}|& f_i.|x_i-\bar{x}|\\ 47-49& 48& 2& 96& 6,45& 12,49\\ 50-52& 51& 4& 204& 3,45& 13,8\\ 53-55& \text{54}& 6& \text{324}& \text{0,45}& \text{2,7}\\ 56-58& 57& 5& 285& 2,55& 12,75\\ 59-61& 60& 3& 180& 5,55& 16,65\\ \text{Jumlah}& & 20& 1089& & 58,8\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \begin{array}{rl}\bar{x}& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.x_i}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & =54+{\displaystyle \frac{1089}{20}=54+0,45=54,45}\end{array}\\ & \begin{array}{rl}SR& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{58,8}{20}}\\ & =2,94\end{array}\\ & \text{Jadi, simpangan rata-ratanya adalah}SR=2,94\end{array}\\ \\ & \mathbf{\text{Alternatif 2}}\\ & \text{Perhatikan tabel berikut}\\ & \begin{array}{|ccccccc|}\hline \text{Nilai}& x_i& f_i& d_i& f_i.d_i& |x_i-\bar{x}|& f_i.|x_i-\bar{x}|\\ \text{47-49}& 48& 2& -6& -12& 6,45& 12,49\\ \text{50-52}& 51& 4& -3& -12& 3,45& 13,8\\ \text{53-55}& \text{54}& 6& \text{0}& \text{0}& \text{0,45}& \text{2,7}\\ \text{56-58}& 57& 5& 3& 15& 2,55& 12,75\\ \text{59-61}& 60& 3& 6& 18& 5,55& 16,65\\ \text{Jumlah}& & 20& & 9& & 58,8\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \begin{array}{rl}\bar{x}& =x_s+{\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.d_i}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & =54+{\displaystyle \frac{9}{20}=54+0,45=54,45}\end{array}\\ & \begin{array}{rl}SR& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{58,8}{20}}\\ & =2,94\end{array}\\ & \text{Jadi, simpangan rata-ratanya adalah}SR=2,94\end{array}$.

$\begin{array}{ll}2.& \text{Tentukanlah nilai varian/ragamnya}\\ & \text{dari data soal no.1 di atas}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan tabel berikut}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& x_i& f_i& |x_i-\bar{x}|& (x_i-\bar{x}{)}^2& f_i.(x_i-\bar{x}{)}^2\\ 47-49& 48& 2& 6,45& 41,6025& 83,205\\ 50-52& 51& 4& 3,45& 11,9025& 47,61\\ 53-55& 54& 6& 324& 0,2025& 1,215\\ 56-58& 57& 5& 285& 6,5025& 32,5125\\ 59-61& 60& 3& 180& 30,8025& 92,4075\\ \text{Jumlah}& & 20& & & 256,95\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \text{dan}\bar{x}=54,45(\text{lihat soal no.1})\\ & \text{maka}\\ & \begin{array}{rl}{S}^2& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.(x_i-\bar{x}{)}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{256,95}{20}}\\ & =12,8475\end{array}\\ & \text{Jadi, varian/ragamnya adalah}\\ & S^2=12,8475\end{array}$.

$\begin{array}{ll}3.& \text{Tentukanlah nilai simpangan baku dari}\\ & \text{dari data soal no.1 di atas}\\ \\ & \mathbf{\text{Jawab}}:\\ & S=\sqrt{S^2}=\sqrt{12,8475}\approx 3,58\end{array}$.

Ukuran Penyebaran Data Tunggal (Materi Kelas XII Matematika Wajib) (Bagian 1)

A. Pengertian

Ukuran penyebaran data adalah nilai dari ukuran yang memberikan gambaran sejauh mana data menyebar atau menyimpang (dispersi/deviasi) dari ukuran pemusatan data. Dalam hal ini bagian yang akan disinggung dalam materi ini adalah: Jangkauan (Range), Jangkauan antar kuartil, Simpangan kuartil, Simpangan rata-rata, Ragam (Variansi), Simpangan baku (Deviasi Standar), Koefisien variansi.

$\begin{array}{|clc|}\hline \text{No}& \text{Data Dispersi}& \text{Simbol}\\ 1.& \text{Jangkauan}& R\text{atau}J\\ 2.& \text{Jangkauan}& H\\ & \text{antarkuartil}& \\ 3.& \text{Simpangan}& Q_d\\ & \text{kuartil}& \\ 4.& \text{Langkah}& L\\ 5.& \text{Pagar dalam}& Q_1-L\\ 6.& \text{Pagar luar}& Q_3-L\\ 7.& \text{Simpangan}& SR\\ & \text{rata-rata}& \\ 8.& \text{Ragam/variansi}& S^2\\ 9& \text{Simpangan baku}& S\\ 10.& \text{Koefisien variansi}& V\\ \hline\end{array}$.

Sebagai catatan bahwa $H$ selain disebut jangkauan antarkuartil sebagaian ada yang menyebut dengan istilah rentang antar kuartil dan terkadang pula dengan sebutan jangkauan interkuartil (Inter Quartile Range) dan juga terkadang menyebutnya dengan hamparan. Untuk $Q_d$  selanjutnyanya ada yang buku yang menyebutnya dengan istilah simpangan kuartil terkadang juga rentang semi interkuartil atau jangkauan antarkuartil.

Perhatikan gambar distribusi frekuensi suatu data berikut

B. Ukuran Penyebaran Data

B. 1 Data Tunggal

$\begin{array}{|clc|}\hline \text{No}& \text{Data}& \text{Formula}\\ 1.& R\text{atau}J& x_{max}-x_{min}\\ 2.& H& Q_3-Q_1\\ 3.& Q_d& {\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ 4.& L& {\displaystyle \frac{3}{2}(Q_3-Q_1)}\\ 5.& Q_1-L& Q_1-L\\ 6.& Q_3-L& Q_3-L\\ 7.& SR& {\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ 8.& S^2& {\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}\\ 9& S& \sqrt{S2}\\ 10.& V& {\displaystyle \frac{S}{\bar{x}}\times 100\mathrm{\%}}\\ \hline\end{array}$.

Catata: Data ukuran yang kurang dari pagar dalam dan atau lebih besar dari pagar luar dinamakan pencilan.

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Diberikan data berikut}\\ & \begin{array}{llllll}30& 32& 32& 43& 50& 51\\ 53& 53& 58& 58& 58& 60\\ 63& 64& 66& 67& 68& 69\\ 70& 72& 75& 78& 80& 82\\ 84& 85& 86& 86& 83& 83\end{array}\\ & \text{Tentukan}\\ & \text{a}.\text{Jangkauan}\\ & \text{b}.Q_1,Q_2,\text{dan}{Q}_3\\ & \text{c}.\text{Jangkauan Antarkuartil}\\ & \text{d}.\text{Simpangan Kuartil}\\ & \text{e}.\text{Pagar Dalam}\\ & \text{f}.\text{Pagar Luar}\\ & \text{g}.\text{Pencilan}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan sajian data dalam bentuk}\\ & \text{diagram}\mathbf{\text{batang daun}}\text{berikut}\\ & \begin{array}{|cl|}\hline \mathbf{\text{Batang}}& \mathbf{\text{Daun}}\\ 3& 022\\ 4& 3\\ 5& 0133888\\ 6& 0346789\\ 7& 0258\\ 8& 02334566\\ \hline\end{array}\\ & \begin{array}{rl}\text{Diketa}& \text{hui}n=30\\ \text{a}.J& =x_{max}-x_{min}=86-30=56\\ \text{b}.Q_1& =\left(x_{{.}_{\frac{1}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{1}{4}.30+\frac{1}{2}}}\right)=x_{.8}=53\\ Q_2& =\left(x_{{.}_{\frac{2}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{2}{4}.30+\frac{1}{2}}}\right)\\ & ={\displaystyle \frac{x_{.15}+x_{.16}}{2}={\displaystyle \frac{66+67}{2}=66,7}}\\ Q_3& =\left(x_{{.}_{\frac{3}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{3}{4}.30+\frac{1}{2}}}\right)=x_{.23}=80\\ \text{c}.H& =Q_3-Q_1\\ & =x_{{.}_{23}}-x_{{.}_8}=80-53=27\end{array}\end{array}$

$.\begin{array}{rl}\text{d}.Q_d& ={\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ & ={\displaystyle \frac{1}{2}(H)={\displaystyle \frac{1}{2}\left(27\right)=13,5}}\\ \text{e}.L& ={\displaystyle \frac{3}{2}(H)={\displaystyle \frac{3}{2}(27)=40,5}}\\ \text{P}& \text{agar dalam}:\\ & =Q_1-L=53-40,5=12,5\\ \text{P}& \text{agar luar}:\\ & =Q_1-L=80+40,5=120,5\\ \text{g}.\text{D}& \text{ari fakta yang ada data ukuran}\\ & \text{yang besarnya kurang dari}\\ & \text{pagar dalam dan lebih besar dari}\\ & \text{pagar luar tidak ada, maka}\\ & \text{tidak ada}\mathbf{\text{data pencilan}}\end{array}$.

$\begin{array}{ll}2.& \text{Diberikan data berikut}\\ & \begin{array}{llllll}73& 74& 66& 65& 68& 65\\ 60& 64& 78& 79& 81& 61\\ 72& 74& 71& 68& 75& 76\\ 96& 56& 64& 80& 84& 43\end{array}\\ & \text{Tentukan}\\ & \text{a}.\text{Jangkauan}\\ & \text{b}.Q_1,Q_2,\text{dan}{Q}_3\\ & \text{c}.\text{Jangkauan Antarkuartil}\\ & \text{d}.\text{Simpangan Kuartil}\\ & \text{e}.\text{Pagar Dalam}\\ & \text{f}.\text{Pagar Luar}\\ & \text{g}.\text{Pencilan}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan sajian data dalam bentuk}\\ & \text{diagram}\mathbf{\text{batang daun}}\text{berikut}\\ & \begin{array}{|cl|}\hline \mathbf{\text{Batang}}& \mathbf{\text{Daun}}\\ 4& 3\\ 5& 6\\ 6& 014455688\\ 7& 123445689\\ 8& 013\\ 9& 6\\ \hline\end{array}\\ & \begin{array}{rl}\text{Diketa}& \text{hui}n=24\\ \text{a}.J& =x_{max}-x_{min}=96-43=53\\ \text{b}.Q_1& =\left(x_{{.}_{\frac{1}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{1}{4}.24+\frac{1}{2}}}\right)=x_{{}_{6,5}}\\ & ={\displaystyle \frac{1}{2}(x_{{.}_6}+x_{{.}_7})={\displaystyle \frac{64+65}{2}=64,5}}\\ Q_2& =\left(x_{{.}_{\frac{2}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{2}{4}.24+\frac{1}{2}}}\right)=x_{{}_{12,5}}\\ & ={\displaystyle \frac{x_{.12}+x_{.13}}{2}={\displaystyle \frac{71+72}{2}=71,5}}\\ Q_3& =\left(x_{{.}_{\frac{3}{4}n+\frac{1}{2}}}\right)=\left(x_{{.}_{\frac{3}{4}.24+\frac{1}{2}}}\right)=x_{{}_{18,5}}\\ & ={\displaystyle \frac{x_{{}_{18}}+x_{{}_{19}}}{2}={\displaystyle \frac{76+78}{2}=77}}\\ \text{c}.H& =Q_3-Q_1\\ & =77-64,5=12,5\end{array}\end{array}$.

$.\begin{array}{rl}\text{d}.Q_d& ={\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ & ={\displaystyle \frac{1}{2}(H)={\displaystyle \frac{1}{2}(12,5)=6,26}}\\ \text{e}.L& ={\displaystyle \frac{3}{2}(H)={\displaystyle \frac{3}{2}(12,5)=18,75}}\\ \text{P}& \text{agar dalam}:\\ & =Q_1-L=64,5-18,75=45,75\\ \text{P}& \text{agar luar}:\\ & =Q_1-L=77+18,75=95,75\\ \text{g}.\text{D}& \text{ari fakta di atas terdapat}\mathbf{\text{pencilan}}\\ & \text{yaitu}:43\text{dan}96\end{array}$.

$\begin{array}{ll}3.& \text{Diberikan data berikut}\\ & \begin{array}{llllll}\text{a}.& 3& 4& 5& 6& 7\\ \text{b}.& 1& 2& 5& 8& 9\end{array}\\ & \text{Tentukan}\\ & \text{a}.\text{Simpangan rata-rata}\\ & \text{b}.\text{Ragam}\\ & \text{c}.\text{Simpangan baku}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Untuk data}:3,4,5,6,7\\ & \begin{array}{rl}\text{Diketahu}& \text{i}n=5\\ \text{a}.\bar{x}=& {\displaystyle \frac{3+4+5+6+7}{5}=\frac{25}{5}=5}\\ \text{sel}& \text{anjutnya}\\ \text{SR}& ={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & ={\displaystyle \frac{1}{5}(|3-5|+|4-5|+|5-5|+|6-5|+|7-5|)}\\ & ={\displaystyle \frac{1}{5}(|-2|+|-1|+\left|0\right|+\left|1\right|+\left|2\right|)}\\ & ={\displaystyle \frac{1}{5}(2+1+0+1+2)}\\ & ={\displaystyle \frac{6}{5}=1,2}\end{array}\\ & \begin{array}{rl}\text{b}.{\text{S}}^2& ={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}\\ & ={\displaystyle \frac{1}{5}((3-5{)}^2+(4-5{)}^2+(5-5{)}^2+(6-5{)}^2+(7-5{)}^2)}\\ & ={\displaystyle \frac{1}{5}(4+1+0+1+4)}\\ & ={\displaystyle \frac{1}{5}(8)}\\ & ={\displaystyle \frac{8}{5}=1,6}\\ \text{c}.S& =\sqrt{S^2}\\ & =\sqrt{1,6}\approx 1,26\end{array}\\ \\ & \text{Dan untuk data}:1,2,5,8,9\\ & \begin{array}{rl}\text{Diketahu}& \text{i}n=5\\ \text{a}.\bar{x}=& {\displaystyle \frac{1+2+5+8+9}{5}=\frac{25}{5}=5}\\ \text{sel}& \text{anjutnya}\\ \text{SR}& ={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n|x_i-\bar{x}|}}\\ & ={\displaystyle \frac{1}{5}(|1-5|+|2-5|+|5-5|+|8-5|+|9-5|)}\\ & ={\displaystyle \frac{1}{5}(|-4|+|-3|+\left|0\right|+\left|3\right|+\left|4\right|)}\\ & ={\displaystyle \frac{1}{5}(4+3+0+3+4)}\\ & ={\displaystyle \frac{14}{5}=2,8}\end{array}\\ & \begin{array}{rl}\text{b}.{\text{S}}^2& ={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{(x_i-\bar{x})}^2}}\\ & ={\displaystyle \frac{1}{5}((1-5{)}^2+(2-5{)}^2+(5-5{)}^2+(8-5{)}^2+(9-5{)}^2)}\\ & ={\displaystyle \frac{1}{5}(16+9+0+9+16)}\\ & ={\displaystyle \frac{1}{5}(50)}\\ & ={\displaystyle \frac{50}{5}=10}\\ \text{c}.S& =\sqrt{S^2}\\ & =\sqrt{10}\approx 3,16\end{array}\end{array}$.

$\text{LATIHAN SOAL}$.

$\begin{array}{ll}1.& \text{Tentukan nilai Jangkauan},Q_1,Q_2,Q_3\\ & hamparan,\text{simpangan kuartil, langkah}\\ & \text{pagar dalam, pagar luar, dan pencilan}\\ & \text{dari data berikut}\\ & \begin{array}{ll}\text{a}.& 3,5,7,9,1,2,8,2,3,4,3,5,7\\ \text{b}.& 10,11,12,13,8,9,4,5,7,5\end{array}\end{array}$.

$\begin{array}{ll}2.& \text{Tentukan simpangan rata-rata}\\ & \text{ragam, dan simpangan baku}\\ & \text{dari data berikut}\\ & \begin{array}{ll}\text{a}.& 3,5,7,9,1\\ \text{b}.& 10,11,12,13,8,9,4,15,7,5\end{array}\end{array}$.

$\begin{array}{ll}3.& \text{Empat buah bilangan memiliki mean,}\\ & \text{tentukanlah keempat bilangan tersebut}\end{array}$.

$\begin{array}{ll}4.& \text{Diketahui datum-datum}\\ & :x-4,x-2,x+1,x+2,x+4,x+5\\ & \text{tentukanlah}\\ & \text{a}.\text{nilai simpangan baku(nyatakan dalam): }x\\ & \text{b}.\text{nilai}x\text{dan simpangan baku jika mean}\\ & \text{dari data di atas adalah 9}\end{array}$.

$\begin{array}{ll}5.& \text{Diketahui simpangan baku}\\ & :2,4,7,11,9-n,9+n\text{adalah}\sqrt{11}\\ & \text{tentukanlah}\\ & \text{a}.\text{mean}\\ & \text{b}.\text{nilai}n\text{yang mungkin}\end{array}$.

DAFTAR PUSTAKA

1. Johanes, Kastolan, Sulasim. 2004. Kompetensi Matematika SMA Kelas 2 Semester 1 Program Ilmu Alam Kurikulum Berbasis Kompetensi. Jakarta: YUDHISTIRA.
2. Kanginan, M., Terzalgi, Y. 2014. Matematika untuk SMA-MA/SMK Kelas XI (Wajib). Bandung: Srikandi Empat Widya Utama.
3. Sharma, S.N., dkk. 2017. Jelajah Matematika 3 SMA Kelas XII Program Wajib. Jakarta: YUDHISTIRA.

Interpolasi Linear

Sumber ada di sini

materi pendukung untuk desil klik di sini dan

materi persentil klik di sini.

Interpolasi linear adalah sebuah metode yang digunakan untuk penentuan titik di antara dua buah titik yang sudah diketahui dan segaris.

Perhatikanlah ilustrasi gambar berikut

dengan proses seperti menentukan persamaan garis lurus diperoleh rumus:

$\begin{array}{rl}{\displaystyle \frac{y-y_0}{y_1-y_0}}& =\frac{x-x_0}{x_1-x_0}\\ y-y_0& =\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukan taksiran nilai dari}\\ & \text{a}.\sqrt{5}\text{c}.\sqrt{12}\\ & \text{b}.\sqrt{7}\text{d}.\sqrt{22}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{a}.\text{Di}& \text{ketahui bahwa}\\ & \{\begin{array}{ll}\sqrt{4}& =2\\ \sqrt{5}& =?\\ \sqrt{9}& =3\end{array}\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ & \approx 2+{\displaystyle \frac{5-4}{9-4}(3-2)}\\ & \approx 2+{\displaystyle \frac{1}{5}}\\ & \approx 2+0,2\\ & \approx 2,2\end{array}\\ & \begin{array}{rl}\text{b}.\text{Di}& \text{ketahui bahwa}\\ & \{\begin{array}{ll}\sqrt{4}& =2\\ \sqrt{7}& =?\\ \sqrt{9}& =3\end{array}\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ & \approx 2+{\displaystyle \frac{7-4}{9-4}(3-2)}\\ & \approx 2+{\displaystyle \frac{3}{5}}\\ & \approx 2+0,6\\ & \approx 2,6\end{array}\\ & \begin{array}{rl}\text{c}.\text{Di}& \text{ketahui bahwa}\\ & \{\begin{array}{ll}\sqrt{9}& =3\\ \sqrt{12}& =?\\ \sqrt{16}& =4\end{array}\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ & \approx 3+{\displaystyle \frac{12-9}{16-9}(4-3)}\\ & \approx 3+{\displaystyle \frac{3}{7}}\\ & \approx 3+0,43\\ & \approx 3,43\end{array}\\ & \begin{array}{rl}\text{d}.\text{Di}& \text{ketahui bahwa}\\ & \{\begin{array}{ll}\sqrt{16}& =4\\ \sqrt{22}& =?\\ \sqrt{25}& =5\end{array}\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ & \approx 4+{\displaystyle \frac{22-16}{25-16}(5-4)}\\ & \approx 4+{\displaystyle \frac{6}{9}}\\ & \approx 4+0,67\\ & \approx 4,67\end{array}\end{array}$.

$\begin{array}{ll}2.& \text{Diberikan data berikut berkaitan dengan}\\ & \text{penduduk di suatu daerah A}\\ & \begin{array}{|cll|}\hline \text{Tahun}& 2015& 2020\\ \begin{array}{rl}& \text{Jumlah jiwa}\\ & \text{daerah A}\end{array}& 340.000& 600.000\\ \hline\end{array}\\ & \text{Tentukan perkiraan jumlah penduduk}\\ & \text{daerah A saat tahun 2018}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}\text{Di}& \text{ketahui bahwa}\\ & \{\begin{array}{ll}\sqrt{2015}& =340.000\\ \sqrt{2018}& =?\\ \sqrt{2020}& =600.000\end{array}\\ y& =y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)\\ & \approx 340.000+{\displaystyle \frac{(2018-2015)}{(2020-2015)}(600.000-340.000)}\\ & \approx 340.000+{\displaystyle \frac{3}{5}\left(260.000\right)}\\ & \approx 340.000+156.000\\ & \approx 496.000\end{array}\end{array}$.

Ukuran Letak Data (Materi Kelas XII Matematika Wajib) Bagian 3

 D. Persentil

$\begin{array}{rl}& \text{Dilambangkan dengan}{P}_i\\ & \text{dibaca: persentil ke}-i\\ & \text{Rumus data tunggal}:\\ & \mathbf{\text{Dengan rumus pendekatan interpolasi linear}}\\ & P_i=\text{datum ke}-{\displaystyle \frac{i}{100}(n+1)}\\ & \text{jika}{\displaystyle \frac{i}{100}(n+1)\text{tidak bulat, gunakan rumus}}\\ & \text{interpolasi linear, yaitu}:\\ & P_i=x_k+d(x_{k+1}-x_k)\\ & \text{dengan}d\text{adalah nilai desimalnya}\\ & \mathbf{\text{Dengan tanpa rumus interpolasi linear}}\\ & \begin{array}{|cc|}\hline n\text{ganjil}& n\text{genap}\\ P_1=x_{{.}_{\frac{1}{100}(n+1)}}& P_1={\displaystyle \frac{1}{2}(x_{{.}_{\frac{1}{100}n}}+x_{{.}_{\frac{1}{100}n+1}})}\\ P_2=x_{{.}_{\frac{2}{100}(n+1)}}& P_2={\displaystyle \frac{1}{2}(x_{{.}_{\frac{2}{100}n}}+x_{{.}_{\frac{2}{100}n+1}})}\\ ⋮& ⋮\\ P_{99}=x_{{.}_{\frac{99}{100}(n+1)}}& P_{99}={\displaystyle \frac{1}{2}(x_{{.}_{\frac{99}{100}n}}+x_{{.}_{\frac{99}{100}n+1}})}\\ \hline\end{array}\\ & \text{Catatan: sesuaikan dengan kondisi soal}\\ & \text{Rumus data berkelompok/berfrekuensi}:\\ & P_i=L_i+\left({\displaystyle \frac{{\displaystyle \frac{i}{100}n-f_k}}{f}}\right)\times c\\ & \text{Penjelasan}\\ & \begin{array}{rl}{D}_i& =\text{persentil ke}-i\\ i& =1,2,3\\ L_i& =\text{tepi bawah kelas persentil ke}-i\\ f_k& =\text{frekuensi kumulatif sebelum}\\ & \text{sebelum kelas persentil ke}-i\\ f& =\text{frekuensi kelas persentil ke}-i\\ c& =\text{panjang kelas interval}\\ n& =\text{banyak data/kelas interval}\end{array}\end{array}$.

Catatan :untuk bahasan interpolasi linear ada di sini

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukan persentil ke-29 dan ke-75 dari data berikut}\\ & 4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ \\ & \text{Jawab}:\\ & \text{Banyak datum}=15\\ & \mathbf{\text{dengan rumus pendekatan interpolasi linear}}\\ & \text{Data mula-mula}\\ & :4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ & \text{Data setelah diurutkan}\\ & :2,3,4,4,4,5,5,6,6,6,7,7,8,8,9\\ & \begin{array}{rl}{P}_{29}& ={\displaystyle \frac{29}{100}(15+1)}\\ & ={\displaystyle \frac{464}{100}=4,64}\\ & =x_4+0,64(x_5-x_4)\\ & =4+0,64(5-5)=4+0=4\\ P_{75}& ={\displaystyle \frac{75}{100}(15+1)}\\ & ={\displaystyle \frac{1200}{100}=12}\\ & =x_{12}\\ & =7\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Persentil ke-32}\left(P_{32}\right)\text{dari data berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Nilai}& f\\ 41-45& 7\\ 46-50& 12\\ 51-55& 9\\ 56-60& 8\\ 61-65& 4\\ \hline\end{array}\\ & \begin{array}{l}\\ \text{a}.& 46\\ \text{b}.& 47\\ \text{c}.& 48\\ \text{d}.& 51\\ \text{e}.& 52\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Diketahui}& \text{persentil ke}-32=P_{32},\\ & \text{dengan}n=\sum f=40\\ P_i& =L_i+c\left({\displaystyle \frac{{\displaystyle \frac{i\times n}{100}-f_k}}{f}}\right)\\ P_{32}& =\text{datum ke}-\left({\displaystyle \frac{32n}{100}}\right)\\ & =x_{\frac{32\times 40}{100}}=x_{12,8}\\ \text{Dan}{x}_{12}& \text{terletak di kelas interval}:46-50\\ P_{32}& =545,5+5\left({\displaystyle \frac{12,8-7}{12}}\right)\\ & =45,5+0,48333...\\ & =45.9833...\approx 46\end{array}\end{array}$

Ukuran Letak Data (Materi Kelas XII Matematika Wajib) Bagian 2

 C. Desil

$\begin{array}{rl}& \text{Dilambangkan dengan}{D}_i\\ & \text{dibaca: desil ke}-i\\ & \text{Rumus data tunggal}:\\ & \mathbf{\text{Dengan rumus pendekatan interpolasi linear}}\\ & D_i=\text{datum ke}-{\displaystyle \frac{i}{10}(n+1)}\\ & \text{jika}{\displaystyle \frac{i}{10}(n+1)\text{tidak bulat, gunakan rumus}}\\ & \text{interpolasi linear, yaitu}:\\ & D_i=x_k+d(x_{k+1}-x_k)\\ & \text{dengan}d\text{adalah nilai desimalnya}\\ & \mathbf{\text{Dengan tanpa rumus interpolasi linear}}\\ & \begin{array}{|cc|}\hline n\text{ganjil}& n\text{genap}\\ D_1=x_{{.}_{\frac{1}{10}(n+1)}}& D_1={\displaystyle \frac{1}{2}(x_{{.}_{\frac{1}{10}n}}+x_{{.}_{\frac{1}{10}n+1}})}\\ D_2=x_{{.}_{\frac{2}{10}(n+1)}}& D_2={\displaystyle \frac{1}{2}(x_{{.}_{\frac{2}{10}n}}+x_{{.}_{\frac{2}{10}n+1}})}\\ ⋮& ⋮\\ D_9=x_{{.}_{\frac{9}{10}(n+1)}}& D_9={\displaystyle \frac{1}{2}(x_{{.}_{\frac{9}{10}n}}+x_{{.}_{\frac{9}{10}n+1}})}\\ \hline\end{array}\\ & \text{Catatan: sesuaikan dengan kondisi soal}\\ & \text{Rumus data berkelompok/berfrekuensi}:\\ & D_i=L_i+\left({\displaystyle \frac{{\displaystyle \frac{i}{10}n-f_k}}{f}}\right)\times c\\ & \text{Penjelasan}\\ & \begin{array}{rl}{D}_i& =\text{desil ke}-i\\ i& =1,2,3\\ L_i& =\text{tepi bawah kelas desil ke}-i\\ f_k& =\text{frekuensi kumulatif sebelum}\\ & \text{sebelum kelas desil ke}-i\\ f& =\text{frekuensi kelas desil ke}-i\\ c& =\text{panjang kelas interval}\\ n& =\text{banyak data/kelas interval}\end{array}\end{array}$.

Catatan :untuk bahasan interpolasi linear ada di sini

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukanlah}{D}_1,D_2,D_3,D_4,D_5,D_6\\ & D_7,D_8,D_9\text{dari data berikut}\\ & 2,3,8,9,2,4,5,8,9\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Total datum}=9\\ & \text{Data mula-mula}:2,3,8,9,2,4,5,8,9\\ & \text{Setelah data diurutkan menjadi}\\ & :2,2,3,4,5,8,8,9,9\\ & D_i={\displaystyle \frac{i}{10}(n+1).\text{Jika hasilnay tidak bulat}}\\ & \text{maka dihitung dengan}{D}_i=x_k+d.(x_{k+1}-x_k)\\ & \text{Sehingga}\\ & \begin{array}{rl}{D}_1& ={\displaystyle \frac{1}{10}(9+1)=\frac{10}{10}=1}\\ & =x_1=2\\ D_2& ={\displaystyle \frac{2}{10}(9+1)=\frac{20}{10}=2}\\ & =x_2=2\\ D_3& ={\displaystyle \frac{3}{10}(9+1)=\frac{30}{10}=3}\\ & =x_3=3\\ D_4& ={\displaystyle \frac{4}{10}(9+1)=\frac{40}{10}=4}\\ & =x_4=4\\ D_5& ={\displaystyle \frac{5}{10}(9+1)=\frac{50}{10}=5}\\ & =x_5=5\\ D_6& ={\displaystyle \frac{6}{10}(9+1)=\frac{60}{10}=6}\\ & =x_6=8\\ D_7& ={\displaystyle \frac{7}{10}(9+1)=\frac{70}{10}=7}\\ & =x_7=8\\ D_8& ={\displaystyle \frac{8}{10}(9+1)=\frac{80}{10}=8}\\ & =x_8=9\\ D_9& ={\displaystyle \frac{9}{10}(9+1)=\frac{90}{10}=9}\\ & =x_9=9\end{array}\end{array}\end{array}$.

$\begin{array}{ll}2.& \text{Tentukan desil ke-4 dan ke-6 dari data berikut}\\ & 4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ \\ & \text{Jawab}:\\ & \text{Banyak datum}=15\\ & \mathbf{\text{dengan rumus pendekatan interpolasi linear}}\\ & \text{Data mula-mula}\\ & :4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ & \text{Data setelah diurutkan}\\ & :2,3,4,4,4,5,5,6,6,6,7,7,8,8,9\\ & \begin{array}{rl}{D}_4& ={\displaystyle \frac{4}{10}(15+1)}\\ & ={\displaystyle \frac{64}{10}=6,4}\\ & =x_6+0,4(x_7-x_6)\\ & =5+0,4(5-5)=5\\ D_6& ={\displaystyle \frac{6}{10}(15+1)}\\ & ={\displaystyle \frac{96}{10}=9,6}\\ & =x_9+0,6(x_{10}-x_9)\\ & =6+0,6(6-6)=6\end{array}\end{array}$.

$\begin{array}{l}\\ 3.& \text{Desil ke-8}\left(D_8\right)\text{dari data berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Nilai}& f\\ 41-45& 7\\ 46-50& 12\\ 51-55& 9\\ 56-60& 8\\ 61-65& 4\\ \hline\end{array}\\ & \begin{array}{l}\\ \text{a}.& 58\\ \text{b}.& 57,5\\ \text{c}.& 57\\ \text{d}.& 56,75\\ \text{e}.& 56,25\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Diketahui}& \text{desil ke}-8=D_8,\text{dengan}n=\sum f=40\\ D_i& =L_i+c\left({\displaystyle \frac{{\displaystyle \frac{i\times n}{10}-f_k}}{f}}\right)\\ D_8& =\text{datum ke}-\left({\displaystyle \frac{8n}{10}}\right)=x_{\frac{8\times 40}{10}}=x_{32}\\ \text{Dan}{x}_{32}& \text{terletak di kelas interval}:56-60\\ D_8& =55,5+5\left({\displaystyle \frac{32-28}{8}}\right)\\ & =55,5+2,5\\ & =58\end{array}\end{array}$

Ukuran Letak Data (Materi Kelas XII Matematika Wajib) Bagian 1

A. Pendahuluan

Sebelumnya telah dipelajari tentang salah satu bentuk ukuran pemusatan data yaitu median yang membagi sebuah data menjadi dua bagian yang sama. Selain median ada juga istilah lain yang dapat membagi sebuah data menjadi beberapa bagian yang sama pula, yaitu kuartl yang membagi sebuah data menjadi 4 bagian yang sama. Kemudian selain kuartil, ada juga desil yang memabgi sebuah data menjadi 10 bagian yang sama serta persentil yang membagi sebuah data menjadi 100 bagian yang sama pula.

B. Kuartil

$\begin{array}{rl}& \text{Dilambangkan dengan}{Q}_i\\ & \text{dibaca: kuartil ke}-i\\ & \text{Rumus data tunggal}:\\ & \mathbf{\text{Dengan rumus pendekatan interpolasi linear}}\\ & Q_i=\text{datum ke}-{\displaystyle \frac{i}{4}(n+1)}\\ & \text{jika}{\displaystyle \frac{i}{4}(n+1)\text{tidak bulat, gunakan rumus}}\\ & \text{interpolasi linear, yaitu}:\\ & Q_i=x_k+d(x_{k+1}-x_k)\\ & \text{dengan}d\text{adalah nilai desimalnya}\\ & \mathbf{\text{Dengan tanpa rumus interpolasi linear}}\\ & \begin{array}{|cc|}\hline n\text{ganjil}& n\text{genap}\\ Q_1=x_{{.}_{\frac{1}{4}(n+1)}}& Q_1=x_{{.}_{\frac{1}{4}n+\frac{1}{2}}}\\ Q_2=x_{{.}_{\frac{2}{4}(n+1)}}& Q_2=x_{{.}_{\frac{2}{4}n+\frac{1}{2}}}\\ Q_3=x_{{.}_{\frac{3}{4}(n+1)}}& Q_3=x_{{.}_{\frac{3}{4}n+\frac{1}{2}}}\\ \hline\end{array}\\ & \text{Catatan: sesuaikan dengan kondisi soal}\\ & \text{Rumus data berkelompok/berfrekuensi}:\\ & Q_i=L_i+\left({\displaystyle \frac{{\displaystyle \frac{i}{4}n-f_k}}{f}}\right)\times c\\ & \text{Penjelasan}\\ & \begin{array}{rl}{Q}_i& =\text{kuartil ke}-i\\ i& =1,2,3\\ L_i& =\text{tepi bawah kelas kuartil ke}-i\\ f_k& =\text{frekuensi kumulatif sebelum}\\ & \text{sebelum kelas kuartil ke}-i\\ f& =\text{frekuensi kelas kuartil ke}-i\\ c& =\text{panjang kelas interval}\\ n& =\text{banyak data/kelas interval}\end{array}\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukanlah}{Q}_1,Q_2,Q_3\text{dari data}\\ & \text{berikut}\\ & \text{a}.3,5,7,1,2,4,9,7\\ & \text{b}.2,3,8,9,2,4,5,8,9\\ \\ & \text{Jawab}:\\ & \mathbf{\text{Dengan rumus pendekatan interpolasi linear}}\\ & \begin{array}{rl}\text{a}.& \text{Total datum}=8\\ & \text{Data mula-mula}:3,5,7,1,2,4,9,7\\ & \text{Setelah data diurutkan menjadi}\\ & :1,2,3,4,5,7,7,9\\ & Q_i={\displaystyle \frac{i}{4}(n+1)\{\begin{array}{ll}{Q}_1& ={\displaystyle \frac{1}{4}(8+1)=2{\displaystyle \frac{1}{4}}}\\ \\ Q_2& ={\displaystyle \frac{2}{4}(8+1)=4{\displaystyle \frac{1}{2}}}\\ \\ Q_3& ={\displaystyle \frac{3}{4}(8+1)=6{\displaystyle \frac{3}{4}}}\end{array}}\\ & Q_1=x_2+{\displaystyle \frac{1}{4}(x_3-x_2)=2+{\displaystyle \frac{1}{4}=2{\displaystyle \frac{1}{4}}}}\\ & Q_2=x_4+{\displaystyle \frac{1}{2}(x_5-x_4)=4+{\displaystyle \frac{1}{2}=4{\displaystyle \frac{1}{2}}}}\\ & Q_3=x_6+{\displaystyle \frac{3}{4}(x_7-x_6)=7+0=7}\end{array}\\ & \begin{array}{rl}\text{b}.& \text{Total datum}=9\\ & \text{Data mula-mula}:2,3,8,9,2,4,5,8,9\\ & \text{Setelah data diurutkan menjadi}\\ & :2,2,3,4,5,8,8,9,9\\ & Q_i={\displaystyle \frac{i}{4}(n+1)\{\begin{array}{ll}{Q}_1& ={\displaystyle \frac{1}{4}(9+1)=2{\displaystyle \frac{1}{2}}}\\ \\ Q_2& ={\displaystyle \frac{2}{4}(9+1)=5}\\ \\ Q_3& ={\displaystyle \frac{3}{4}(9+1)=7{\displaystyle \frac{1}{2}}}\end{array}}\\ & Q_1=x_2+{\displaystyle \frac{1}{2}(x_3-x_2)=2+{\displaystyle \frac{1}{1}=2{\displaystyle \frac{1}{2}}}}\\ & Q_2=x_5=5\\ & Q_3=x_7+{\displaystyle \frac{1}{2}(x_8-x_7)=8+{\displaystyle \frac{1}{2}=8{\displaystyle \frac{1}{2}}}}\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Data penjualan suatu barang setiap bulan}\\ & \text{di sebuah toko pada tahun 2019 adalah}:\\ & 20,3,9,11,4,12,1,9,9,12,8,10.\\ & \text{Median, kuartil bawah, dan kuartil atasnya}\\ & \text{berturut-turut adalah}....\\ & \begin{array}{l}\\ \text{a}.& 6{\displaystyle \frac{1}{2},3\frac{1}{2},\text{dan}9\frac{1}{2}}\\ \text{b}.& 9,6,\text{dan}11{\displaystyle \frac{1}{2}}\\ \text{c}.& 6{\displaystyle \frac{1}{2},9,\text{dan}12}\\ \text{d}.& 9,4,\text{dan}12\\ \text{e}.& 9,3{\displaystyle \frac{1}{2},\text{dan}12}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \mathbf{\text{Dengan tanpa rumus interpolasi linear}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}\\ :& 20,3,9,11,4,12,1,9,9,12,8,10\\ \text{Sete}& \text{lah data diurutkan}\\ :& 1,3,4,8,9,9,9,10,11,12,12,20\\ \text{Dike}& \text{tahui}n=12\mathbf{\text{genap}}\\ Q_1& =x_{\frac{1}{4}n+\frac{1}{2}}=x_{\frac{1}{4}.12+\frac{1}{2}}=x_{3,5}=6\\ Q_2& =x_{\frac{2}{4}n+\frac{1}{2}}=x_{\frac{2}{4}.12+\frac{1}{2}}=x_{6,5}=9=M_e\\ Q_3& =x_{\frac{3}{4}n+\frac{1}{2}}=x_{\frac{3}{4}.12+\frac{1}{2}}=x_{9,5}=11{\displaystyle \frac{1}{2}}\\ \text{Sela}& \text{njutnya data dapat dituliskan}\\ & 1,3,\underset{\begin{array}{c}\Downarrow \\ Q_1\end{array}}{\underbrace{4,8}},9,\underset{\begin{array}{c}\Downarrow \\ Q_2=M_e\end{array}}{\underbrace{9,9}},10,\underset{\begin{array}{c}\Downarrow \\ Q_3\end{array}}{\underbrace{11,12}},12,20\end{array}\end{array}$

$\begin{array}{l}\\ 3.& (\mathbf{\text{UN IPA 2014}})\\ & \text{Kuartil atas dari data pada tabel berikut}\\ & \text{adalah}....\\ & \begin{array}{|cc|}\hline \text{Data}& f\\ 20-25& 4\\ 26-31& 6\\ 32-37& 6\\ 38-43& 10\\ 44-49& 12\\ 50-55& 8\\ 56-61& 4\\ \hline\end{array}\\ & \begin{array}{l}\\ \text{a}.& 49,25\\ \text{b}.& 48,75\\ \text{c}.& 48,25\\ \text{d}.& 47,75\\ \text{e}.& 47,25\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}& \text{Kuartil atas}=Q_3,\text{dengan}n=\sum f=50\\ & \text{Kita sertakan lagi tabel di atas berikut}\\ & \begin{array}{|cc|}\hline \text{Data}& f\\ 20-25& 4\\ 26-31& 6\\ 32-37& 6\\ 38-43& 10\\ \text{44-49}& \text{12}\\ 50-55& 8\\ 56-61& 4\\ \hline\end{array}\\ & Q_3=\text{Datum ke}-\left({\displaystyle \frac{3n}{4}}\right)=x_{\frac{3.50}{4}}=x_{37,5}\\ & \text{dan}{x}_{37,5}\text{terletak di kelas interval}44-49\\ & Q_3=L_3+c\left({\displaystyle \frac{{\displaystyle \frac{3n}{4}-f_k}}{f}}\right)\\ & =43,5+6\left({\displaystyle \frac{37,5-26}{12}}\right)\\ & =43,5+{\displaystyle \frac{11,5}{2}}\\ & =49,5+5,75\\ & =49,25\end{array}\end{array}$